Localization and number of visited valleys for a transient diffusion in random environment

We consider a transient diffusion in a $(-\kappa \mathrm{/}2)$-drifted Brownian potential $W\mathrm{\_}\kappa$ with $0\text{textless}\kappa \text{textless}1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h\mathrm{\_}t$-minima of the environment, for $h\mathrm{\_}t$ a bit smaller than $\log t$. We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on a decomposition of the trajectory of $W\mathrm{\_}\kappa$ in the neighborhood of $h\mathrm{\_}t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W\mathrm{\_}\kappa$ and of $W\mathrm{\_}\kappa$ Doob-conditioned to stay positive.